wp_list_categories, Add class to all list items with children

jQuery solution:

You could try this if you want to use jQuery:

<script>
jQuery(document).ready(function($) {
    $('li.cat-item:has(ul.children)').addClass('i-have-kids');
}); 
</script>

to add the class i-have-kids to all the li parents that include the items ul.children, within the HTML generated from wp_list_categories().

Category walker solution:

You could take a look at the Walker_Category class in /wp-includes/category-template.php and extend it with an extra part like:

$termchildren = get_term_children( $category->term_id, $category->taxonomy );
if(count($termchildren)>0){
    $class .=  ' i-have-kids';
}

If we skip the feed image and feed parts, the extended walker could look like this:

class Walker_Category_Find_Parents extends Walker_Category {
    function start_el( &$output, $category, $depth = 0, $args = array(), $id = 0 ) {
        extract($args);

        $cat_name = esc_attr( $category->name );
        $cat_name = apply_filters( 'list_cats', $cat_name, $category );
        $link = '<a href="' . esc_url( get_term_link($category) ) . '" ';
        if ( $use_desc_for_title == 0 || empty($category->description) )
            $link .= 'title="' . esc_attr( sprintf(__( 'View all posts filed under %s' ), $cat_name) ) . '"';
        else
            $link .= 'title="' . esc_attr( strip_tags( apply_filters( 'category_description', $category->description, $category ) ) ) . '"';
            $link .= '>';
            $link .= $cat_name . '</a>';

        if ( !empty($show_count) )
            $link .= ' (' . intval($category->count) . ')';

                if ( 'list' == $args['style'] ) {
                        $output .= "\t<li";
                        $class="cat-item cat-item-" . $category->term_id;

                        $termchildren = get_term_children( $category->term_id, $category->taxonomy );
                        if(count($termchildren)>0){
                            $class .=  ' i-have-kids';
                        }

                        if ( !empty($current_category) ) {
                                $_current_category = get_term( $current_category, $category->taxonomy );
                                if ( $category->term_id == $current_category )
                                        $class .=  ' current-cat';
                                elseif ( $category->term_id == $_current_category->parent )
                                        $class .=  ' current-cat-parent';
                        }
                        $output .=  ' class="' . $class . '"';
                        $output .= ">$link\n";
                } else {
                        $output .= "\t$link<br />\n";
                }
        }
    }

You could further take out the parts that you don’t need.

Usage example:

<?php 
$args = array(
  'taxonomy'     => 'my_custom_taxonomy_slug',
  'orderby'      => 'name',
  'hide_empty'   => 0,
  'title_li'     => '',
  'hierarchical' => 1,
  'walker'       => new Walker_Category_Find_Parents(),
);
?>
<ul class="menu">
   <?php wp_list_categories( $args ); ?>
</ul>

Output example:

Here’s a list example, using the Walker_Category_Find_Parents walker:

with the following HTML structure:

<ul class="menu">
    <li class="cat-item cat-item-1">
        <a href="http://example.com/category/plants/">plants</a>
    </li>
    <li class="cat-item cat-item-2 i-have-kids">
        <a href="http://example.com/category/animals/">animals</a>
        <ul class="children">
            <li class="cat-item cat-item-3 i-have-kids">
                <a href="http://example.com/category/animals/birds/">birds</a>
                <ul class="children">
                    <li class="cat-item cat-item-4">
                        <a href="http://example.com/category/animals/birds/falcons/">falcons</a>
                    </li>
                </ul>
            </li>
        </ul>
    </li>
    <li class="cat-item cat-item-5">
        <a href="http://example.com/category/stones">stones</a>
    </li>
</ul>

I just removed the title attributes to make it more readable.

But you can see where the i-have-kids class is added to the li tags with children.

When I visit the / Animals / Birds / category, the HTML structure becomes:

<ul class="menu">
    <li class="cat-item cat-item-1">
        <a href="http://example.com/category/plants/">plants</a>
    </li>
    <li class="cat-item cat-item-2 i-have-kids current-cat-parent">
        <a href="http://example.com/category/animals/">animals</a>
        <ul class="children">
            <li class="cat-item cat-item-3 i-have-kids current-cat">
                <a href="http://example.com/category/animals/birds/">birds</a>
                <ul class="children">
                    <li class="cat-item cat-item-4">
                        <a href="http://example.com/category/animals/birds/falcons/">falcons</a>
                    </li>
                </ul>
            </li>
        </ul>
    </li>
    <li class="cat-item cat-item-5">
        <a href="http://example.com/category/stones">stones</a>
    </li>
</ul>